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5

$\epsilon$, usually by definition it stands for any number greater than 0, so it represents numbers arbitrarily close to 0.
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GraphthJan 11 '12 at 17:01

3

@Graphth: this is similar to saying that the largest number used in a proof is $N$, which is often allowed to tend to infinity. Neither $\epsilon$ nor $N$ stand for particular numbers; but particular numbers, I believe, are what Andreas is asking for.
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Niel de BeaudrapJan 11 '12 at 17:05

I think this question is currently too vague. A better formulation might be: "What is the smallest constant that has explicitly appeared in a published paper?"
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Grumpy ParsnipJan 11 '12 at 18:06

2

From Littlewood's "A Mathematician's Miscellany", p.38: "A minute I wrote (about 1917) for the Ballistic Office ended with the sentence 'Thus $\sigma$ should be made as small as possible'. This did not appear in the printed minute. But P. J. Grigg said, 'what is that?' A speck in a blank space at the end proved to be the tiniest $\sigma$ I have ever seen (the printers must have scoured London for it)."
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Per ManneJul 11 '14 at 13:13

In 1928, A. S. Besicovitch introduced and studied regular and irregular $1$-sets in the plane (a fundamental notion in fractal geometry). He proved that the lower $1$-density of a regular $1$-set $E$ in the plane is equal to $1$ (the maximum possible value) at almost all (Lebesgue measure) points in $E$. To show how differently irregular $1$-sets in the plane behaved, Besicovitch proved that the lower $1$-density of an irregular $1$-set $F$ in the plane is bounded below $1$ at almost all (Lebesgue measure) points in $F$. The bound that Besicovitch obtained in 1928 was $1 - 10^{-2576}$.

In 1934, Besicovitch managed to improve this to $\frac{3}{4}$. [See Section 3.3 of Kenneth J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985.] "Besicovitch's $\frac{1}{2}$-problem" (presently unsolved, I believe) is to find the best almost everywhere upper bound for irregular sets. Besicovitch himself showed by a specific example that this bound is at least $\frac{1}{2},$ and he conjectured that it is equal to $\frac{1}{2}.$ In 1992, David Preiss and Jaroslav Tiser proved the bound is at most $\frac{2 \;+ \;\sqrt{46}}{12},$ which is approximately $0.73186.$ For a lot more about the Besicovitch $\frac{1}{2}$-problem, see Hany M. Farag's papers at